VDG∞: Galaxy Redshift-Space Clustering Model

• VDG∞ is a theoretical framework that models the galaxy two-point correlation function in redshift space using a cumulant expansion and non-Gaussian damping to capture FoG effects.
• The model parameterizes cosmological, bias, and damping effects with EFT counterterms, achieving unbiased recovery of parameters down to 20 h⁻¹ Mpc.
• It employs a fast computational pipeline with the COMET emulator and FFTlog transforms, outperforming alternative models in accuracy and bias control.

The velocity difference generator (VDG), specifically the VDG$_\infty$ model, is a theoretical framework for modeling the galaxy two-point correlation function (2PCF) in redshift space. It employs a cumulant expansion to capture the non-perturbative effects of galaxy peculiar velocities, particularly the non-Gaussian "Finger-of-God" (FoG) damping, enabling accurate cosmological parameter estimation at small spatial separations relevant for Stage-IV surveys such as those conducted by Euclid (Collaboration et al., 8 Jan 2026).

1. Formalism and Mathematical Structure

The VDG$_\infty$ model operates on the redshift-space moment

$$\langle e^{-i k\mu\,f\,[u_z(x)-u_z(x')]} [1+f\,\nabla_z u_z(x)+\delta(x)] [1+f\,\nabla_z u_z(x')+\delta(x')] \rangle$$

where $u_z(x)$ denotes the line-of-sight peculiar velocity at position $x$, $f$ is the linear growth rate, $\delta(x)$ is the density fluctuation, $k$ is the Fourier wavenumber, and $\mu$ is the cosine of the angle to the line-of-sight.

A damping prefactor, $D(k,\mu)$, is factored out to capture non-perturbative FoG effects, and the remainder is expanded to one loop. The final model for the redshift-space galaxy power spectrum has the form

$_\infty$0

where:

• $_\infty$1 are the real-space galaxy and velocity power spectra,
• $_\infty$2 and $_\infty$3 are loop corrections treated with a bias expansion.

The distinctive feature of VDG$_\infty$4 is the non-Gaussian damping kernel,

$_\infty$5

where $_\infty$6, $_\infty$7 is computed from the linear power spectrum,

$_\infty$8

and $_\infty$9 is a free kurtosis parameter encoding deviations from Gaussianity in the pairwise velocity distribution.

To obtain configuration-space multipoles, the model employs the transformations: $$\langle e^{-i k\mu\,f\,[u_z(x)-u_z(x')]} [1+f\,\nabla_z u_z(x)+\delta(x)] [1+f\,\nabla_z u_z(x')+\delta(x')] \rangle$$0 followed by an FFTlog Hankel transform.

2. Parameterization and Model Components

The VDG$$\langle e^{-i k\mu\,f\,[u_z(x)-u_z(x')]} [1+f\,\nabla_z u_z(x)+\delta(x)] [1+f\,\nabla_z u_z(x')+\delta(x')] \rangle$$1 parameterization incorporates cosmological, bias, damping/FoG, and counterterm parameters:

Category	Parameters	Features / Priors
Cosmology / Linear Power	Fixed templates: $$\langle e^{-i k\mu\,f\,[u_z(x)-u_z(x')]} [1+f\,\nabla_z u_z(x)+\delta(x)] [1+f\,\nabla_z u_z(x')+\delta(x')] \rangle$$2; Full-shape: $$\langle e^{-i k\mu\,f\,[u_z(x)-u_z(x')]} [1+f\,\nabla_z u_z(x)+\delta(x)] [1+f\,\nabla_z u_z(x')+\delta(x')] \rangle$$3 (optionally $$\langle e^{-i k\mu\,f\,[u_z(x)-u_z(x')]} [1+f\,\nabla_z u_z(x)+\delta(x)] [1+f\,\nabla_z u_z(x')+\delta(x')] \rangle$$4)	Vary per snapshot
Galaxy Bias (EFT)	Eulerian $$\langle e^{-i k\mu\,f\,[u_z(x)-u_z(x')]} [1+f\,\nabla_z u_z(x)+\delta(x)] [1+f\,\nabla_z u_z(x')+\delta(x')] \rangle$$5; Non-local $$\langle e^{-i k\mu\,f\,[u_z(x)-u_z(x')]} [1+f\,\nabla_z u_z(x)+\delta(x)] [1+f\,\nabla_z u_z(x')+\delta(x')] \rangle$$6, $$\langle e^{-i k\mu\,f\,[u_z(x)-u_z(x')]} [1+f\,\nabla_z u_z(x)+\delta(x)] [1+f\,\nabla_z u_z(x')+\delta(x')] \rangle$$7	Redshift-independent
FoG / Non-Gaussianity	$$\langle e^{-i k\mu\,f\,[u_z(x)-u_z(x')]} [1+f\,\nabla_z u_z(x)+\delta(x)] [1+f\,\nabla_z u_z(x')+\delta(x')] \rangle$$8 (kurtosis), prior $$\langle e^{-i k\mu\,f\,[u_z(x)-u_z(x')]} [1+f\,\nabla_z u_z(x)+\delta(x)] [1+f\,\nabla_z u_z(x')+\delta(x')] \rangle$$9	Marginalized
EFT Counterterms	$u_z(x)$0, $u_z(x)$1, $u_z(x)$2 (multiply $u_z(x)$3), priors $u_z(x)$4	Redshift-independent

Bias and counterterm priors remain constant across redshifts, while cosmological parameters are allowed to vary with each simulation snapshot.

3. Computational Implementation

The computational pipeline for VDG$u_z(x)$5 utilizes the following elements:

• Power spectra and correction terms ($u_z(x)$6) are generated by the COMET emulator, delivering $u_z(x)$7 plus counterterms in $u_z(x)$8 ms.
• $u_z(x)$9 is built directly in COMET, with $x$0 multipoles $x$1 computed by Gaussian quadrature over $x$2.
• Hankel transforms to configuration space are performed using FFTlog/Talman (as implemented in "hankl"), including a mild Gaussian damping factor $x$3 ($x$4 Mpc$x$5) to suppress ringing, relevant only for $x$6 Mpc.
• Parameter space is explored using PyMultiNest (3000 live points, efficiency 0.8, tolerance 0.5). The likelihood employs an analytic Gaussian covariance matrix iterated five times.

4. Data Sets, Fitting Regime, and Results

The primary dataset consists of Flagship 1 H$x$7 galaxies in a $x$8 Mpc box at redshifts $x$9 with number densities ranging from $f$0 to $f$1 Mpc$f$2. Multipoles $f$3 are measured in $f$4 Mpc bins over $f$5–$f$6 Mpc, with covariances derived analytically.

Fitting is performed under two schemes:

• Template-fitting: Linear power spectrum is fixed; fit parameters include $f$7 and nuisance, for $f$8 in $f$9Mpc.
• Full-shape analysis: Recomputes $\delta(x)$0 at each sample in $\delta(x)$1, includes same nuisance, bias, and counterterms.

Key results at $\delta(x)$2:

• VDG$\delta(x)$3 provides unbiased parameter recovery down to $\delta(x)$4Mpc in both fits, with mean reduced $\delta(x)$5 and figure-of-bias (FoB) $\delta(x)$6.
• Recovered cosmological parameters ($\delta(x)$7) agree with fiducial values to $\delta(x)$8, with precision (FoM) $\delta(x)$9 and $k$0.
• Other models (EFT, CLPT, CLEFT) exhibit substantial bias or loss of accuracy at small $k$1.

5. Comparative Performance with Other Theoretical Frameworks

The VDG$k$2 approach is contrasted with effective field theory (EFT), convolutional Lagrangian perturbation theory (CLPT), and its effective-field extension (CLEFT):

Model	Unbiased Scale (template/full-shape) $k$3 at $k$4	$k$5	FoB	FoM
VDG$k$6	20 / 20	$k$7	1.3	$k$8
EFT	30 / 25–30	$k$9	3.2	–
CLPT	$\mu$0 / $\mu$1	1.1	2.9	–
CLEFT	25 / 20	1.04	1.7	–

In both template and full-shape fits, only VDG$\mu$2 successfully yields unbiased cosmological inferences to $\mu$3Mpc for the lowest redshift snapshot. This performance is matched only by CLEFT in the full-shape analysis.

6. Strengths, Limitations, and Ongoing Challenges

The VDG$\mu$4 framework possesses several notable advantages:

• The non-Gaussian damping $\mu$5 models FoG effects more accurately than Gaussian or Lorentzian EFT kernels.
• EFT counterterms are integrated to control ultraviolet sensitivity.
• Enables unbiased recovery of both growth and geometric parameters at the smallest scales tested ($\mu$6 down to $\mu$7 Mpc).
• Achieves comparable or superior figures of merit to other models, despite more nuisance parameters.

Principal limitations and caveats:

• The method requires the COMET emulator for computational feasibility; direct perturbation theory is significantly slower.
• The kurtosis parameter $\mu$8 is phenomenological, lacking a first-principles determination and requiring marginalization.
• There is an ongoing need to test VDG$\mu$9 against more realistic synthetic catalogs, including light-cone effects, redshift errors, and lensing magnification.
• In the presence of degeneracies (e.g., $D(k,\mu)$0 with bias in full-shape fits), supplementing VDG$D(k,\mu)$1 with complementary statistics such as the bispectrum may be beneficial.

7. Role in Configuration-Space Analysis for Cosmological Surveys

For configuration-space analyses of the 2PCF multipoles $D(k,\mu)$2 targeting precision cosmology with data from Euclid and similar spectroscopic galaxy surveys, VDG$D(k,\mu)$3 is identified as the baseline model. It uniquely enables exploitation of small-scale data ($D(k,\mu)$4 Mpc) without introducing bias in inferred cosmological parameters. CLEFT is recommended as a cross-check within this context, with future work foreseen in extending these models to incorporate survey-specific systematics and more complex observational effects (Collaboration et al., 8 Jan 2026).